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THE DECOMPOSITION INTO CELLS OF THE AFFINE WEYL GROUPS OF TYPE A
J-Y SHI
Thesis submitted for the degree of Doctor of Philosophy at Warwick University.
April, 1984 Mathematics Institute
ACKNOWLEDGEMENT
First and foremost, Z wish to thank my supervisor. Professor R.W. Carter, who has been a constant source of encouragement.and ideas, and who has helped me over my
difficulties. Further, I am indebted to Professor G. Lusztig, who permitted me to use his recent unpublished results on the cells of the affine Weyl groups, by which the results of my thesis are made more satisfactory. I am also grateful to Ms Terri Moss who has so patiently and cheerfully done the difficult job of typing.
I wish also to acknowledge the financial support of the Chinese government and the Committee of Vice-Chancellors and Principals of the universities of the United Kingdom.
In [1], Kazhdan and Lusztig introduce the concept of a W-graph for a Coxeter group W. In particular, they define left, right and two-sided cells. These W-graphs play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated. These cells have been worked out only in a very few cases. In the present thesis, we shall find all the left, right and two-sided cells in the affine Weyl group An of type A n > 2. Our main results show that each left (resp. right)
n-i
cell of determines a partition, say X of n and, is characterized by a X-tabloid and also by its generalized right (resp. left )
T-invariant. There exists a one-to-one correspondence between
the set of two-sided cells of An and the Bet An of partitions of n. The number of left (resp. right) cells corresponding to a given partition X € AR is equal to mn *-- , where
n y . I
j-1 *
y - {y1 > ... > ym ) is the dual partition of X. Each two- sided cell in An is also an RL-equivalence class of AR and is a connected set. Each left (resp. right) cell in An is a maximal left (resp. right) connected component in the two-sided cell of An containing it. Let P be any proper standard parabolic subgroup of AR isomorphic to the symmetric group Sn< Then the intersection of V with each two-sided cell of AR is non-empty and is just a two-sided cell of t. The intersection of V with each left (rasp, right) cell of An is either empty or a left (resp. right) cell of AR .
C O N T E N T S
Abstract
CHAPTER 1 t INTRODUCTION
§1.1 Cells and some of their elementary properties §1.2 A statement of our main results
CHAPTER 2 l ELEMENTARY PROPERTIES OF THE AFFINE WEYL GROUP A OF TYPE A ,, n > 2
n n— i
§2.1 Three descriptions of the affine Weyl group Ar
§2.2 The functions &(w), £ (w), R(w) on the affine Weyl group Ar
§2.3 The subsets ®L tst ), ® R (st ) of the affine Weyl group AR , n > 3
§2.4 Some terminology
CHAPTER 3 > THE PARTITION OF n ASSOCIATED WITH AN ELEMENT OF THE AFFINE WEYL GROUP A_n
CHAPTER 4 t SOME CELLS OF THE AFFINE WEYL GROUP k n CHAPTER 5 t ITERATED STAR OPERATIONS AND INTERCHANGING
OPERATIONS ON BLOCKS §5.1 Iterated star operations
§5.2 Some results on Iterated star operations §5.3 The .Interchanging operations P . 1 and 0 . 2
\a2 a1
§5.4 More general Interchanging operations
CHAPTER 6 t THE SUBSET o“ 1 (X) OF THE AFFINE WEYL GROUP AR §6.1 Two simple lemmas on Iterated star operations 16.2 The subset F of the affine Weyl group AR 16.3 The subset H^ of o- 1 (X)
§6.4 o~1 (X) is a union of RL-equlvalenoe classes
(1 )
n n
S8.1 Definition of Ï 89
n
mm
§8.2 The map r\t k k^ 91
n n
§8.3 The partition associated with an element of & n 91
§8.4 The functions i(w), £(w), R(w) and star operations
in Ï 92
n
§8.5 Interchanging operations on blocks in Ar 94
§8.6 Totally ordered sets with a distance function 98
am
§8.7 Deletion operations in in 103
§8.8 Commutativity of interchanging operations w i t h
deletion 104
§8.9 Commutativity of interchanging operations w ith
the map n '• 107
CHAPTER 9 < THE SEQUENCE Ç(w,k) BEGINNING WITH AN ELEMENT
OF » x 109
§9.1 A description of 109
§9.2 A sequence Ç(w,r) beginning with an element of H x 110
§9.3 The deletion map d(X,m) 113
§9.4 The subset ^ of S’ 1 ( M 116
9 mm mm f*
§9.5 The sequence £(w,k) beginning with w € 126
§9.6 The sequence £(w,k) beginning with w € > x 130
§9.7 Increasing chains 131
C H A P T E R 10 < R A I S I N G O P E R A T I O N S ON L A Y E R S 156
§1 0 . 1
§1 0 . 2
§10.3 §10.4
CHAPTER
§1 1 . 1 §1 1 . 2 §11.3 §11.4 CHAPTER CHAPTER §13.1 §13.2 CHAPTER CHAPTER 115.1 §15.2 Reflective pairs
Raising operations on layers
Proof of proposition 10.2.3 when 1 < u < Xr Proof of Proposition 10.2.3 when Xk+1 < u < xk and 1 < k < r
11 t THE LEFT AND RIGHT CELLS IN a” 1 (X) The map T fro m to the set of X-tabloids The set S o f principal normalized elements
The subset of
The number o f left cells in o~1 (X)
12 s o~1 (X) IS AN RL-EQUIVALENCE CLASS OF A n 13 t LEFT CELLS ARE CHARACTERIZED BY THE
GENERALIZED RIGHT T"INVARIANT
156 162 166 174 182 182 185 192 19 4 202 20 3 Left cells are characterized by the generalized *
right t-invariant 2Q5
The standard parabolic subgroup ?n 20 5
14 t THE TWO-SIDED CELLS OF THE AFFINE WEYL GROUP
A_ 215
n
15 > SOME PROPERTIES OF CELLS AND OTHER EQUIVALENCE
CLASSES OF A_ 221
n
The commutativity between a left star operation
and a right star operation 221
Connectness of cells and other equivalence classes
of A„ 22 5
n
We shall first define cells of any given Coxeter group and state some of their elementary properties, most of which appear In [1], and then Introduce our work.
1.1 CELLS AND SOME OF THEIR ELEMENTARY PROPERTIES
Let W = <W,S> be a Coxeter group with S the set of Coxeter •*
generators. There Is an associative algebra H over the polynomial ring X[q] with basis elements {Tw |w e W} satisfying the following relations:
TwTw' “ Tww' if * i(w) + A(w')
(Ts+1) (Tg-q) - 0, if s £ S,
here &(w) is the length of w. We define the Hecke algebra K to be K ©s jgjA, where A - Z[g^,g- ^].
To construct a representation of H endowed with a special basis, we define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional datat for each vertex x e X, we are given a subset Xx of S and, for each ordered pair of vertices y, x such that (y,x) € Y, we are given an integer ti(y,x) t
These data are subject to the following requirements! let E be the free A-module with basis X. Then for any s € S,
-x, if s e i
t, (x) -
{
qx ♦ q* I y(y,x)y, if s € I
y€X *
SCXy
(y^c)€V
defines an endomorphism of E (i.e. the sum. over y is assumed to be always finite) and there is a unique representation
<t>:K *■ End(E) such that $(Ts ) = xs for each s € S.
We shall construct, for any W, such a graph. First, we give some definitions. Let a -*■ a be the involution of the ring A defined by q* = q - ^. This extends to an involution h ■+ h of the ring H, defined by
zaw Tw " z5wT~-1
(Note that Ttf is an invertible element H, for example, if s € S, we have T ~ 1 ■ q-1T s ♦ (q_1-1)). For any w € W, we define
qw - q*’^ , ew ■ (-1)*'^W ^. Let < be the Bruhat order relation on W [defined in [9]]: We can now state
Theorem 1.1.1 For any w € W, there is a unique element € W such that
I
j£<w ey cw <*w «y Py,w T
where P is a polynomial in g of degree < i (& (w)- & (y)-1) for
~ y 9*
y < y. «aft pw,w
U.-It is well known that for each y < w with £(w)-t(y) < 2, we have P„ __ ■ 1. y,w In the case that W is either an ordinal Weyl group or an affine Weyl group, Lusxtig [1] [4] has proved that all coefficients of the polynomial P
defines an endomorphism of E (l.e. the sum over y Is assumed to be always finite) and there is a unique representation
<p:U -*■ End (E) such that <ji(Ts ) = t s for each s € S.
We shall construct, for any W, such a graph. First, we give some definitions. Let a ■> a be the involution of the ring A defined by q* = q ~ ^ . This extends to am involution h ■+• h of the ring H, defined by
2awTw - ^ w t; -i
(Note that Tw is am invertible element H, for exaunple, if s € S, we have T ~ 1 ■> q-1T g + (q_1-1)). For any w € W, we define
qw * q*'^W ^, ew ■ (-1)i'*W ^. Let < be the Bruhat order relation on W [defined in [9]] : We can now state
Theorem 1.1.1 For any w € W, there is a unique element € H such that
where P is a polynomial in q of degree < 1 (&(w)- & (y)-1) for " y fw
y < w and Pu u - 1 .
It is well known that for each y < w with t(w)-i(y) < 2, we have P ■ 1. in the case that W is either an ordinal Weyl
Jr 9 ™
group or an affine Weyl group, Lusstig [1] [4] has proved that
all coefficients of the polynomial P are non-negative integers.
-3
Definition 1.1.2 Given y, w € W, we say that y ^ w if the following conditions are satisfiedt y < w, ew = -ey and Py w
(given by Theorem 1.1.1) is a polynomial in g of degree exactly J(i.(w)- i(y)-1). In this case, the leading coefficient of P
y
t
"is denoted by y(y,w). It is a non-zero integer. If w ■( y, we set y(w,y) = y(y,w).
By this definition, it is easily seen that for any y < w with i(w) = A(y) + 1, we have y 4, w.
Let W° be the group opposed to W. Then (W x w°, S J_L s °) is
a Coxeter group. Let rw be the graph with its vertex set {w|w € W} and its edge set {{y,w}|y ^ w}. For each w 6 W, let Iw * £ (w) _LJ_ *(w)° c s JJ_ S°, where £ (w) * {s € S|sw < w>, R(w) = {s £ S|ws < w).
Theorem 1.1.3 rw , together with the assignment w Iw and with the function y defined above, is a W x w°-graph.
Now given any W-graph r, and a subset S' c s, we can regard
r as W'-graph (where W' is the subgroup of W generated by S') by replacing the set Ix e S for each vertex x of r# by the set
Ix n S'. In particular, rw can be regarded as a W-graph and as
a W°-graph.
regarded as a full subgraph o f r (with the same sets Ix and
the same function y) is itself a W-graph. The set of equivalence classes is an ordered set w ith respect to In the case of the W x w°-graph rw , the equivalence classes for ^ are called the
2-sided cells of W. When rw is regarded as a W-graph, we shall
use the notation L instead of ^ ; the corresponding
equivalence classes are called the left cells of W. When rw is
regarded as ,a W°-graph, we shall use the notation instead
of pi the corresponding equivalence classes are called the right cells of W. A minimal non-empty set which is both a union of left cells and a union of right cells is called a RL-equivalence class, written w ^ y If w,y lie in the same RL-equivalence class. Clearly, any 2-sided cell is a union of some RL-equivalence classes
We now state a property of the preorders ^ and ^ on W.
Theorem A [1, Proposition 2.4]
(i) If x ^ y, then R(x) => 8 (y) . Hence, if x £ y, then B(x) ■ R(y)
(ii) If x j y, then £ (x) = £ (y). Hence, if x £ y, then £ (x) - £(y)
Let us fix two generators s,t in S such that st has order 3.
Let
If w € PT (s,t ) , then exactly one of the elements sw, tw is in L
Dt (s,t), we denote it *w. The map w + *w is an involution of li
DL (s,t). Similarly, we have an involution w + w* of PR (s,t) : w* is the unique element of PR (s,t) n {ws,wt}. Let <s,t> be the group of order 6 generated by s,t. We have
Theorem B [1, Theorem 4.2]
Let y,w be two elements in PL (s,t).
(i) If yw~1 t <s,t>, then we have y w if and only if *y *w, and then yi(y,w) - u(*y,*w).
(ii) If yw~1 e <s,t>, then we have y •< w if and only if *w •< * y , and then yi(y,w) ° u(*w,»y) - 1 .
Let y tw be two elements in PR (sft ) .
(iii) If y ~ 1w t <s,t>, then we have y ^ w if and only if y* -4 w * , and then u(y,w) - M(y*,w*).
(iv) If y"\( € <s,t>. then we have y A w if and only if w* •< y * . and then y(y,w) ■ w(w»,y*) ■ 1. □
Theorem C [1, Corollary 4.3]
(i) Let y, w be two elements in PL (s,t). If y R w, then *y R *w.
(ii) Let y,w be two elements in PR (s,t). If y £ w, then y* £ w*.
We now define an equivalence relation PL on W generated by w £ *w in PL (s,t) for some s,t € S with st having order 3. We
L
in H which is both a union of PL~equivalence classes and a union of PR-equivalence classes.
Theorem D . For any y, w e W.
- f\j
(i) If w y, then w y.
PL___________ L (ii) If w J'* y, then w £ y.
PR R
(iii) If w p y, then w y.
Proof: (i) cam be reduced to the case when y = *w in DL (s,t) for some s,t € S with st having order 3. Then either y ^ w or w y. Since the sets £ (y) fl {s,t} and £ (w) n {s,t} both contain exactlv one element and these elements are distinct, this implies
£ (y) ft £(w) and £(w) £ £ (y). So y L w. Similarly for (ii), and (iii) follows from (i), (ii). o
Now we define the generalized right (resp. left) -["invariant of w € W. [8]
Definition 1.1.4 We say w, y € W are equivalent to order zero, £
written w m y, if B(w) - R(y). Inductively, we define equivalence o
to order n for n > 1. We sav w,y are equivalent to order n #
r R
written w M y, if w w y and for every s,t € S with st having R
order 3 and y,w € D„(s,t), we have y' m w', where y' - y* and
R n-1
w' ■ w* in DR (s^t). We say w,y have the same generalized right 2i
T-invariant if w m y for any n > 0. Similarly, we can define the n
§1.2 A STATEMENT OF OUR MAIN RESULTS
From the above statement, we see that the cells play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated when the order of H gets larger. They have been worked out only in a very few cases. For
example we know [5] that in the symmetric groups Sn , n > 1, each left (resp. right) cell determines a partition of n. There exists a one-to-one correspondence between the set of two-sided cells of SR and the set An of partitions of n. In the present thesis, we shall extend these results to the case of the affine Weyl group AR of type AR_ ^ , n > 2.
The affine Weyl group Ar can be described in several different ways: It cam be described as a Coxeter group, as a subgroup of the permutation group on S or a group of affine matrices of period n
(The precise descriptions of all these will be given in Chapter 2) etc. The last two descriptions lead us to define a map
a»A„ -*• \ by Curtis Greene's procedure [7].
n n
Our main results show that for any X .■ {X1 > ... > Xr ) € An the fibre q~1 (X) is a two-sided cell and also an RL-equlvalence class of A . o” 1 (X) consists of ” *■ ■ ■ ■ left (resp. right) cells --- s--- ; v.i
---j-i 3
of An , where y - (u1 > ... » ym ) is the dual partition of X. o_1 (X) is also a connected set (for the definitions of a connected set as well as a maximal left connected component below, see f 15.2).
any t, 1 < t < A... Each left cell is also characterized by its generalized right x-invariant. Let P be any proper standard parabolic subgroup of An isomorphic to the symmetric group Sn . Then the intersection of t with any two-sided cell of An is non empty and is just a two-sided cell of P. The intersection of f with any left (resp. right) cell of An is either empty or a left
(resp. right) cell of P.
In the present thesis, we mainly regard &n as a group of affine matrices of period n.
In Chapter 2, we give three different descriptions of the affine Weyl group An and some elementary properties for this group. We also define a star operation on an element of An which is one of the most important operations in our thesis.
We define a map o:*n •* An In Chapter 3 by Curtis Green's procedure [7] and show that the fibres of o are invariant under star operations.
In Chapter 4, we determine some special cells of An , n > 2. In particular, we get all cells of A2 « So in the subsequent chapters, we shall always assume n > 2.
We define, in Chapter 5, iterated star operations on an element w of An and interchanging operations on blocks of w.
They are all successions of left star operations. These operations are very important in the proof of our main results.
-9
First we show in Chapters 6 and 7 that for any x € An » -1
o (X) is a union of RL-equivalence classes of AR and for any — 1
w € a (X)» there exists an element y of K with y p w, where
A Jj
is the set of left normalized elements (often simply called normalized elements) of a 1 (X).
The second step is the most difficult part in the whole of our proof and includes Chapters 8-11. In this step, we define a map T: N, C for some fixed x C A and then show that for
X 11
w if and only if T(y) = T(w). The proof of this
y, w € y L
result is achieved bv considering a new kind of operation on an
element w of called a raising operation on a layer of w, which
is not in general a succession of star operations but gives an element in the same left cell as w. Since |C^| = — -n- with
)
;in v .i
j-1 3 y = {y^ > ... > yn ) the dual partition of this confirms
Lusztig's first conjecture [2] which says that o_ 1 (x) consists of ---- left (resp. right) cells of An for any x € An .
n yi )
j-T 3
Thirdly, we show in Chapter 12 that o~1 (X) i» just a single RL-eguivalence class of An for any x € An *
Theorem E. Let W be any affine Weyl group. Let z, z' be
__________________ a ...
two elements of Vi which satisfy the following conditionss (i) Either z' > z or z* < z.
(ii) B(z') t R(z) and f(z') t £ (z).
Then z and z ’ are not in the same two-sided, cell of W . o Cl On the other hand, in Chapter 13, we give another character ization of any left cell of Ar in terms of generalized right x-invariant and conclude that any two elements y, w of Ar lie
in the same left cell if and only if they have the same generalized right x-invariant. Let P be any standard parabolic subgroup
of An isomorphic to the symmetric group Sn . We show in Chapter 13 and 14 that the intersection of P with a“ 1 for any X € An is non-empty and is just a two-sided cell of P and then verifies Lusztig's third conjecture in the case W ft ■ An . This conjecture
[3] says that each two-sided cell in W fi meets some proper standard parabolic subgroup of W . where W is any affine Weyl group. We also show that the intersection of P with any left
(resp. right) cell of An is either empty or a left (resp. right) cell of P.
In the last chapter, we show that left star operations commute , with right star operations on an element of An . We also show
that each two-sided cell or P-equivalence class in AR is a connected set and that each left (resp. right) cell or PL~(resp. P_-)equivalence olass in A_ is a left (resp. right) connected set.
K XI
1 1
-CHAPTER 2 : ELEMENTARY PROPERTIES OF THE AFFINE WEYL GROUP A OF TYPE A_ n > 2
n __ n — I ____
In Chapter 1, we have given the definitions of left, right and 2-sided cells as well as RL-equivalence class for an arbitrary Coxeter group. From now on, we shall restrict our attention only to the affine Weyl group Ar of type An _^, n > 2, and consider how to decompose this group into these equivalence classes.
In this chapter, we shall first give three equivalent descriptions of An and then show some basic properties of the length function A(w) and the sets £ (w), R(w) for any w € An . In §2.3, we shall describe the left (resp. right) star operations on w in terms of permutations on X and in terms of affine matrices when w lies in some st+1 ^ <resP* DR (st ,st+1)), where
sfc, st+1 are two Coxeter generators of Afi with Bt8t + ^ having order 3. The star operation is one of the most fundamental
operations throughout our thesis. Finally, in §2.4, we list some terminology on an affine matrix for later use.
§2.1 THREE DESCRIPTIONS OF THE AFFINE WEYL GROUP A, n
An can be described in the following three different ways, (i) By generators and relations
A_ - <s. 11 < i < n
n i 1
m, «
, (■¿■j) ■ 1# for 1 < j < n>
where
“ij •
if i j
if I
t
J, j ± 1with i ♦ Ï to be the natural map from Z to the set of the residue classes n = {T,2,...,n}. We denote A = < i < n}.
(ii) Regarded as a set of pernujtations on Z.
(i+n)w ■ (i)w + n for i € Z"
< wtZ
z
n n L£ (t)w ■ i t
0 t=l t=1
The relation between these two descriptions is as follows: For any i, 1 < i < n, corresponds the permutation
if t
/
I, i + 1if t » i + 1
if t = I
for t 6 Z.
(iii) Regarded as the set A^ of all ® x ~ affine matrices w of type An-1 which are defined as follows:
(a) The integer set Z is the set parametrising its rows (resp. columns). The integers parametrising its rows (resp. columns) are monotone increasing from top to bottom (resp. from left to right).
(b) The entries in each of its rows (resp. columns) are all zero except for one which is 1.
(c) Let (e(u,ju )|u e Z) be the set of its non-zero entries, where e(u,ju ) lies in its (u,ju )-position. Then ju+n ■ ju ♦ n
n n
for any u € S and z - £ u.
u-1 u u-1
1 3
-of its non-zero entry sets {e(u,j ) |u £ S} with S c X, |S| = n and S = {T,5,...,n}, where S is the image of S under the natural map i - I.
For any X, Y £ A£ with (ex (i,j) |i,j € X} and {eY Ci *, j*> |i',j' as their entry sets, let Z = X*Y be an » x « matrix satisfying condition (a) with its (i,j)-entry Z e (i,k)ev (k,j) for any
k£X x *
i, j € X. It is easily shown that Z € A^ and A^ is a group with such multiplication.
For w £ A', we define an <» x «> matrix X.. which satisfies
n w
condition (a), (b) with {e(t,(t)w)|t € X} as its non-zero entries. Then X.. £ A". So we can define a map from A' to A" by w + Xw . It is obvious that such a map is a group isomorphism, where 1 < i < n, corresponds X^ £ A£ with its non-zero
entries {e„ (u,j ) |u € X} satisfying
x i u
u if u i* I, i + 1
*u u— 1 if u ■ I T T
u+1 if u * I.
From now on, we shall identify An with A^ and A^, and denote them all by AR . The set A * { s ^ l < i < n) always denotes its (distinguished) Coxeter generators. We stipulate that ■i+qn ■ for any q, i € X.
12.2 THE FUNCTIONS t(w), £ (w), U(w) ON THE AFFINE WEYL GROUP Afl Let us list some simple properties of A .
Lemma 2.2,1 For y € An , s A £ A,
€ X }
(ii) w = y-s^ is obtained from y by transposing the (i+qn)-th column with the (i+t+qn)-th column for every q € X.
tlii) w = y ~ 1 is obtained from y by transposing y .
Proof t By definition of the affine matrices and their multi
plication. a.
Let ) be the length function of regarded as a Coxeter group. Then we have
2 -.2 -2 &(y> ° i< i;i < n | [ i ^ i ^ H for y € An , where [h] is the integer part of h for h £ ffi.
Proof:
l 1<i<j<n
Let us compare 2 I [-LlLXzJALZ) | with 1<i <j<n n
(j)sty-(i)st (y)
| [--- --- ] | for any 1 < t < n. Since
<h)sty -
this implies that
(h)y if h t {t, t +1} (h+1)y if G ■ t
(h-1)y if E ■ t+1
11— S i --- — 11 - I [(3> V
-When t ^ n, Z |
|{I,3>n{t,S+T)|-i
if (1,3) n {t,t+T> - 0.
|{I,?)n(£+f>|-i
-15-When t = n, 1 < i < n, we assume that (n)y - (i)y * kn ♦ r
and (i)y - (1)y “ k'n + r' with k, k', r,r' € Z and 0 < r, r* < n-1 (actually, 1 < r, r' < n-1 since (n)y + (i)y i (1)y). Then
11 ^ ^ 1 1 = . |(-k')n^(n-r')..] , „ | | . | [JiltillX] |
and
( i l s j - d l s j
It- ■ - V - ^ 1 1 -
|[.<A)r»-.<-n->^]| * |[(-k)n+(n-r)]| = | _k | K ([JiiiiliAiZ] |.
So
wea ls o
have( j ) s _ y - ( i ) s y
2
1 1--- s - s ---
- i1
. 1Ci<j<n
n
|{I,j>n(T,n>|»1
£
I [ UiX-.LUy.i I
1<i<j<n
n
|{I, J >n{ 7, n
} | »1On the other hand, when t + n, we assume that (t+1)y - (t)y *= kn ♦ r with k,r € Z and 1 < r < n.
Then |[(t+ 1)aty~ <t)Bty n . , j i t l x ^ t H i y j | . ,{(,-k-1 lyln-r) ] ,
- I [ (trijyr.jt^ j | * 1 if k s. o
“ l-k-1| - , (1)
[ |[<t*1>y-(t>Y]|- 1 if k < o
(n)s y-(1)s y
II---*-s--- I I
I ^2n+ (1 )y-(n)yj | , (1-h) n+ (n-u),1 n 1
= |1-h|
it(n>y;(1)y ]i ♦ 1
| I - 1
if h < 0
if h > 0
(2)
(j)s y-(i)s y M W - i i W
So Z | [---- ---— ] | - Z | [liLX-lilX] | t 1 (3)
1<i<j<n n 1<i< j<n n
It is easily seen that Z | [-LliXzlilX] | = 0 if and only 1<i<j<n n
if for any 1 < i < j < n, the inequality 1 < (j )y — (i)y < n
holds if and only if y * 1. (4)
Then formulae (3) and (4) imply that i(y) > Z i [.iiLXziilX] I. 1<i<j<n n
To show equality, it suffices to show that for any y i 1, 3 at least one t, 1 < t < n, such that
1<i<j<n II---- *-B---- - I I
i | [ i l l t i l U , | - ,
1<i<j<n n
i.e. to show that at least one of the inequalities (t+1)y - (t)y < 0, for some t, 1 < t < n. (n)y - (1)y > n
holds, or, equivalently, to show that if (t+1)y-(t)y > 0 for all 1 < t < n, then (n)y-(1)y > n. In general, we have
n-1
(n)y-(1)y ■ Z ((t+1 )y-(t)y) > n-1. If (n)y-(1)y ■ n-1, then t»1
n n n-1
equation I t *» Z (t)y = n*(1)y + Z t that (1)y * 1 and so
t-1 t-1 t-1
(t)y = t for all t, 1 < t < n. But this means y = 1. It contradicts y t 1. Also, since (n)y / (1)y, we have
(n)y-(1)y ? n. Thus it follows that (n)y - (1)y > n.
Our assertion is proved. □
Corollary 2.2.3 (of the proof of Lemma 2.2.2) Let w,y € *n , sfc € A with w = sty. Then
(i) M w ) “ M v ) + 1 — » (t+1)y > (t)y (ii) M w ) - M y ) - 1 «■» (t+1)y < (t)y
Prooft We write (t+1)y - (t)y * kn + r with k, r € X and 1 < r < n when t t n. Then by formula (1) and Lemma 2.2.2, l(w) * i(y) ♦ 1 k > 0 ~ (t+1)y-(t)y > 0 « ■ (t+1)y > (t)y.
Also, we write (n)y-(1)y ■ hn + u with h, u € S and 1 < u < n when t ■ n. Then by formula (2) and Lemma 2.2.2,
M w ) - M y ) ♦ 1 h < 0 «— ■ (n)y-(1)y < n «-• (n+1)y-(n)y > 0
«-* (n+1)y > (n)y.
So (i) follows. Since (ii) is equivalent to (i), our
proof is complete. a
Corollary 2.2.3 can be restated in terms of matrices as followsi
Corollary 2. 2 . 3 * i If w is obtained from y by transposing the (i+qn)-th row with the (i*1+qn)-th row for all q € X, and if •y (u»3u ) 1« th* non-zero entry of y lying in the (u,ju )-position for any u C X, then M w ) ■ i (y) ± 1 and
-(i) l(w) = M y ) ♦. 1 ji+1 > j±
(ii) Mw) = M y ) - 1. ji+1 < n
The following two lemmas concern the functions £ ( ), It ( ) on k .n
Lemma 2.2.4 For any w € An , we have
£(w) = {sfc € A|(t+1)w < (t)w> 8 (w) = {st € A|(t+1)w” 1 < (t)w“ 1 }
Proof: By definition of the functions £( ), B( ) (see Chapter
1) and Corollary 2.2.3. a
Lemma 2.2.5 For any w € An , we have 0 < |£(w)| < n and
0 < |8(w)I < n. Moreover, the following three statements are equivalent t
(1) If (w) I - 0 (11) 18 (w) I ■ 0 (111) w ■ 1
Proof» Obviously, w ■ 1 Implies |£(w) | - |B(w) | ■ 0.
In the proof of Lemma 2.2.2, we have shown that w t 1 Implies |£(w)| > 0. And so w + 1 Implies w ~ 1 + 1 and then
|£(w**1)| > 0 . It turns out |8(w)| > 0. Therefore the latter part of the lemma has been proved. Now It Is enough to show that |£(w)| i n. Otherwise, we would have (t)w > (t+1)w for all 1 < t < n. So by Lemma 2.2.4,
(1)w > (2)w > ... > (n)w > (n+1)w - (1)w * n, l.e. n < 0.
Now we assume n > 3. Let us denote PL (st ,st+1) and P R (st ,st+1) by Dl<sfc) and DR (st ), respectively, for any t € *. We call the map w + in (>L (st ) t*le left star operator on w and y y* in DR (st ) the right star operator on y. In terms of permutations on X, w € *n lies in DL (st ) if and only if w satisfies one of the following inequalitiest
(i) (t+1)w < (t)w < (t+2)w (ii) (t+1)w < (t+2)w < (t)w (iii) (t)w < (t+2)w < (t+1)w (iv) (t+2)w < (t)w < (t+1)w. Also, in terms of matrices, w € An lies in C L (s^) if and only if w has one of the following forms:
2.3 T H E S U B S E T S (st > , D R (st > O F T H E A F F I N E W E Y L G R O U P * n , n > 3.
(iii')
)
— ml(ii')
(iv')
where form (o') is the matrix version of inequality (a) for a * i , ii/ iii/ iv.
If w € DL (*t )/ then w, *w (or *w, w) in 0L (*t ) »re either of forms (i')r (ii'), or of forms (iii'), (iv'), respectively.
Since PR (»fc) ■ {w 6 *n |wT € bY I'«®»« 2.2.1 (iii) we have the corresponding results on PR (st ) , where w T is the transpose of the matrix w.
2.3 THE SUBSETS Dr (s.), D.(s.) OF THE AFFINE WEYL GROUP A . n > 3.
^__ w r t n
Now we assume n > 3. Let us denote ^x.^6t'st+1 ^ an<^ DR (st ,sfc+1) by J>L (st ) and t>R <st >» respectively, for any t 6 Z. We call the map w -*■ *w in D T (s. ) the left star operator on w and y y* in pR (st ) the right star operator on y. In terms of permutations on X, w € An lies in DL (sfc) if and only if w satisfies one of the following inequalities«
(i) (t+1)w < (t)w < (t+2)w (ii) (t+1)w < (t+2)w < (t)w (iii) (t)w < (t+2)w < (t+1)w (iv) (t+2)w < (t)w < (t+1)w. Also, in terms of matrices, w € An lies in |>L (st ) if and only if w has one of the following forms«
where form (a') is the matrix version of inequality (a) for o ■ 1» ii, iii, iv.
If w € then w, *w (or *w, w) in 0L (st ) ere either of forms (I'), (ii'), or of forms (iii1)» (iv'), respectively.
(iii') (iv')
Since DR (»t ) ■ {w € An |w € by Lemma 2.2.1(iii)
T
we have the corresponding results on &R (*t)» where w is the transpose of the matrix w.
{w € An |wT € by Lemma 2.2.1(iii)
»Jt J _ • • • 1 . a A tm V •. • Vs a « .a • i t k a
2 0
-S2.4 SOME TERMINOLOGY
Now we make some conventions for later use. Fix an element w € A .
n
(i) When we say "an entry of w", we always mean that this entry is non-zero unless the contrary is indicated. When we mention "an entry of w" (not necessarily non-zero), it means that we know both its value and its position in w. The entries are
usually denoted by e in expressions such as e (w) , e(i,j), e ^ ( i (j), e ( i , (w)), e(w),j), etc, if it lies in the (i,j)-position of w.
(ii) Suppose that E = {e(i^,j 1),...,e(ifc,jfc) } is a subset of entries of w such that i 1 < ... < ifc and j 1 > ... > j^. Then E is called a descending chain of entries of w (briefly, a
descending chain of w ) . Let |E| * t be its size. In that case, we also call {(i.,)w > ... > (it )w> a descending chain of w.
(iii) The submatrix, consisting of any m consecutive rows of w with m < n, is called a block. Blocks will be written as
A, B, C,... . We write |A| - m as the size of A.
(iv) For a block A of w, if e(i+1,j 1),...,e(i+m,j ) are its entries such that i € * and j ^ > ... > jm , than A is called a block of w whose entry set is a descending chain (briefly, a DC block of w ) . In that case, if the entries e(i,jQ ),
local MDC block of w in B. Clearly, any MDC block Is also a local MDC block and any DC block Is a local MDC block In itself.
Assume that A, B are two blocks of w. Then A U B denotes the union of A and B. In particular, when A, B are consecutive
a
blocks of w, we see that [A,B] = (Q ) is a block of w, provided that |A| * |B| < n.
(v) Let e, e' be two entries (not necessarily non-zero) of w. Let f, f' (resp. g, g') be two rows (resp. columns) of w. We define
r(e,e') *
0 if e, e' lie in the same row of w
. ±m otherwise
where m-1 > 0 is the number of rows between e and e'.
r(e,e') = m (resp. r(e,e') - -m) if e* is below (reap, above) e. We also define
c (e,e')
0 if e,e' lie in the same column of w <
11 otherwise
where &-1 > 0 is the number of columns between e and e'. c(e,e') " l (resp. c(e,e') ■ - 1) if s' is on the right (resp. left) of e.
Similarly, we can define r(f,f'), e(g,g'). Clearly,
r(e,e') ■ -r(e',e), c(e,e') ■ -c(e',e), r(e,e”) ■ r(e,e*) ♦ r(e',e”), c(e,e") ■ c(e,e') ♦ c(e',e") for any entry e M of w (not necessarily non-zero), etc.
-22-For any rows f, £' (resp. columns g,g') o£ w, we say that £' is congruent to £ (resp. g 1 is congruent to g) if r ( f , f ) € ni (resp. c(g,g') e nX) .
The set o£ ail entries (resp. rows, columns) congruent to a certain entry (resp. row, coluron) o£ w is called an entry class of w (resp. a row class, a column class). Let ë be a set of m entry classes of w. Then we define the size of ê by |e| = m.
Let ë be a set of entry classes of w. Let ç be the set of ail rows and columns of w each of which contains some entry of ë. Then ç is called a set of row-column classes (briefly, rc-classes) of w. We define the size of ç, written |ç| , by the size of ë.
We cali two blocks A, B of w congruent if the top row of B is congruent to the top row of A and |A| * |B|. For any block A of w with |A| * m, we usually denote any of its congruent blocks also by A. Moreover, let SA - {e(i+u,(w))|1 < u < m} be the set of entries contained in A, SA (q) ■ {e(i+u+qn, (w)) |1 < u < m} for some q € S. Then by abuse of terminology, we also cali SA (q) a block of w and denota it by A.
(vii) For any w € *n , we say that w has the form (A^,...,A^) at i if w has the form
the first row of A^(w) in w. Xf we first say that w has the form
l '
A t (w)
, where A fc(w), 1 < t < l, are consecutive blocks of
t
t
(A^,...rA 1) at i and subsequently mention "the u-th entry of A^. (w) ", it always means that this A fc(w) is a single block of w lying between the (i+1)-th row and the (i+n)-th row. We
denote the u-th entry of A t (w) by e((w), j^(w)), e((w), j^(w,i)) or e((w),j£ (w)). If all A 1 (w),...,A^(w) are DC (resp. MDC)
blocks, then we say that w has the DC (resp. MDC) form (A^,...,A ^ ) at i. If all A 1 (w),...,A£(w) are local MDC blocks in the block
[A& ,...,A^], then we say that w has a local MDC form (AJ^,...,A1) at i. Sometimes, for the sake of emphasizing that the block At (w) of w lies between the (i+1)-th rows and the (i+n)-th row, we denote A^fw) by A^(w,i).
(viii) In the present thesis, for any w € *R , when we say "transposing the i-th and the j-th rows (resp. columns) of w ”, it always means that we transpose the (i+qn)-th and the
(j ♦ qn)-th rows (resp. columns) of w for every q € X. We make the same conventions for the transformations on the blocks of
^ «
w. We also make similar conventions on w € A -n
mt
A will be introduced later). n
-24-CHAPTER 3 t THE PARTITION OF n ASSOCIATED WITH AN ELEMENT OF THE AFFINE WEYL GROUP A_
' ' ■ ■ ■ . . . i i . i n
Let An be the set o£ partitions of n, n > 2. Let A be the set of proper subsets of A. We shall define two maps: o: An -*■ An
the star operations and the inverse for any X € AR . These two maps, especially the first one, are at the heart of our thesis.
In this chapter, we shall also discuss some relations between these two maps.
For any J £ A, Wj is by definition a standard parabolic
subgroup of generated by J.
Definition 3.1 A map n:A -*■ An is defined as follows: For
It is easily seen that the map ir is well defined and surjective.
Definition 3.2 A map o:A_ -*• A_ is defined as follows:
■ ■ .... n n
To w € AR , we associate a sequence of integers d 1 < d 2 < ... < d r ■ n as follows: d^ is the maximum cardinal of a subset of S whose
elements are incogruent to each other mod n and which is a disjoint union of k subsets each of which has its natural order reversed by w. Let ■ d 1# Aj ■ d^ - d^_1 for 1 < j < r. Then it is clear that X, > X, > ... > X, and £ - n. We define
1 * r J
and ir: A -*■ An , and show that the fibre a"*1 (X) is invariant under
J = J, U ...UJ € A with W T - »
i r — j
1 < j < r, and | | > .... ir(J) ■ { |J1 | ♦
t
> |J2 | ♦ 1 > .X ... x W T and W- indecomposable,j ““ “ "j.
r j
we define
> |Jr l + 1 > 1 > ... > 1) € An
The Integers d^, 1 < k < r, in Definition 3.2, can also be described in terms of an affine matrixt dk is the maximum cardinal o f a subset of entries of w whose elements are incon- gruent to each other and which is a disjoint union of k descending chains of w.
The following lemma shows that for any
x
€ An , the fibre (X) is invariant under the inverse.Lemma 3.3 g(w) = o(w~1) for any w € *n
Proof: We know from Lemma 2.2.1 (iii) that the matrix w 1 is the transpose of w. Since the operation of transposing a matrix keeps any descending chain and sends any two incongruent entries to incongruent ones, our result then follows immediately from the definition of the map a. a
For w € An , the following condition on S » 0 ... U St c X is called Cn (w,t)t Elements of S are incongruent mod n and
(a)w > (b)w in Sj, 1 < j < t, implies a < b. Let E * E 1 U ... U Efc be the subset of entries of w with E^ * {a(a,(a)w)|(a)w C S ^ ,
1 < j < t. Then condition Cn (w,t) on S - S 1 U ... U Sfc is equivalent to the following condition on E ■ E^ U ... U E fcs Elements of E are incongruent and E^ is a descending chain for
any 1 < j < t. So we can also say that E > E ^ U ... U Efc satisfies Cn (w,t) and regard condition Cn (w,t) on E ■ E^ U ... U E fc as a matrix version of condition Cn (w,t) on S ■ S 1 U ... U St .
Now we define a preorder > on An as follows! Let X ■ {X1 > X2 > •••> Xr }, v ■ {y1 > > ... > yB > € An . We
2 6
-equivalently, if X1 + X2 ♦ ... + Xk > y 1 ♦ u2 * ••• + Uk for any
1 < k < r. Clearly, this Is a partial order on An . X > p Implies r < m. For any X ={X1 > X2 > ... > Xr > € An , we call
p = {y1 > p 2 > ... > ym > € An the dual partition of X if pk is the number of parts Xj of X with 1 < j < r and Xj > k for any 1 < k < m. The map which sends any element of An to its dual is an involution of An and is order-reversing with respect to >.
Lemma 3.4 For any w, w' € *n and sa € A with w' = s^w and l(w') » i(w) ♦ 1, we have o(w') > o(w).
Proof: If S * S 1 U ... U Sfc e 1 satisfies Cn (w,t) for any t > 1, then S' * (S1)w””1w' U ... U (St )w_1w' satisfies Cn (w',t). This implies that the integer dfc in Definition 3.2 for w' is not less than that for w, for any t > 1. So o(w') > o(w). a
Corollary 3.5 If w' * ws in A_ with s 6 A and i(w') * A(w) ♦ 1,
“ __________ a_____ n a
then a (w1) > o (w) .
Proofi This follows immediately from Lemmas 3.3, 3.4. □
The following lemma gives a relation between two maps a and n on a given element w € A_.
n
Lemma 3.6 For any w € *n , ir(£(w)), tt(R(w)) < o (w).
Proof» By Lemma 3.3, it suffices to show that ir(£(w)) < o(w). Assume that £ (w) ■ J ■ J1 u ... U Jr € A such that
W_ ■ W_ x ... k w_ and W, is indecomposable, 1 < j < r,
U J ^ Wy W j
Then tt (jC (w ) ) = { m . j + 1 > m 2 + 1 > . . . >■ m r + 1 > 1 > . . . > 1 } € A R .
On the other hand, for t with 1 < t < r, let
Sj ■ {(ij + D w , (ij+2)w,..., (ij + n»j + 1 )w } , 1 < j < t.
By Lemma 2.2.4, we have
(ij+1)w > (ij+2)w > ... > (ij + m.. + 1)w, 1 < j «4 t
and this implies that S = S. U ... U S . satisfies C (w,t) with
t t 1 t
ISI ■ E (m.+l). So E (m.+1) < E X . for any t with
j-1 3 j=1 3 j-1 3
1 < t < min {r,h}, where we assume o (w) = {X. > ... > X. }.
t t 1 n
But this easily implies that E (m.+1) < E X. for any t > 1.
j-1 3 j-1 3
Therefore ir(£(w)) < o(w). a
Now we shall show that for any X € An , the fibre o~1 (X) Is invariant under the star operations.
Lemma 3.7 Assume that w' • *w in &-(#.) for some 1 € * Then o ( w 1) - o (w).
Proofi By symmetry, it suffices to discuss the case that
(i+1)w < (i+2)w < (i)w. In that case, w' ■ SjW, i(w') - 4(w)-1 and then for any j € X, we have
r <j)w
i fJ
*
I, ITT<j*1)W
it J
- I
„ <j-1)W
i f5 -
ITTLet S = S 1 U ... U Sfc c * satisfy Cn (w,t). If J 1 < j < t, such that Sj contains two elements (h^)w, (h2 )w with h^ - I and h 2 = h n + 1 , then <S)w"1w' - ( S ^ w ^ w ' U ... U (St )w“ 1w' satisfies C n (w',t). Clearly, | (S)w“ 1w ’ | = |S|. Now suppose that 3 1 < j < t such that contains (h^ w , (h2) w with h 1 = I and h 2 = h 1 + 1. If i (g)w 6 S such that g = i+2, then let
, (S )w-1w' if l / j
s* * < *
* ((h1-*-2)wUSj)w-1w'-(h1)w' if i ■ j
where for «my sets X,Y, X-Y denotes their set-theoretical
difference. If 3(g)w € S such that g - i+2, say (g)w € S^, then it is clear that k / j . Assume that
Sj » { ( i ^ i w > (ij2)w > ... > < ij2 < ••• < i j a ^
Sk “ i(ik 1 )w y (ik 2 ,w > **• > (ikak ) W lik1 < **2 < *** * ik a,t}
-28-and for some 1 < u, u' < aj, 1 < v < ak , we have
ikv - T + T and i^u , ■ i^u +1. Then u' - u+1. By proper choice of S^, we may assume that ikv - i^u + 2 . Then let
(S )w_1w ‘
X if » I1 k,j
{ (ij1)w'>.••>(ij,u-1)w'>(ij»u+1)w'>(ik v )w'>(ik,v+1)w’Lj,u-1
> ...><1kojt)W')lf 1 " j
[j (ik 1 ,w,>*,,>(ik,v-1lw’>(lj,u)w'>(ij,u+2)w,><ij,u+3)w' >.. .><1.. ) W ) if l ■ k.
3 j
An example of this construction of in terms of matrix is given
Sk
— row—
— ij u'H row — row — ik ylii row
s;
/h
—
F l -____ ~ ~ s l 7-vJif;' ✓:::
„ *i/
w'where i. „ - i. „.,-1 = i. -2. w ’ is obtained from w by
J ru J
§
1 * f vtransposing the ij u ~th row with the i^ u + .|-th row.
In both cases, S'
=
u . . . u
satisfies Cn (w',t) and |S'| = |S|. Now for any S = S.j U ... U S ^ c S , t > 1, satisfying Cn (w,t), we can find S' * Sj U ... U c I which satisfiesC n (w',t) with |S'| not less than |S|. So by the same argument as that in the proof of Lemma 3.4, we have o(w') > o(w).
Therefore o(w) = o(w') as required. a
Corollary 3.8 If w' - w* in fl^s^) for some i € I, then
a (w*) - g (w).
Proof: Since w' ■ w* in ®R (®^)• it implies that w'“ 1 ■ *(w-1) in By Lemma 3.7, g(w'” 1 ) - g(w- 1 ). So g(w') ■ g(w) by Lemma 3.3. □
Proposition 3.9 If w, w' € *n satisfy w ^ w * y then g(w) ■ c(w*>.
Proof > Since P-equivalence is generated by w £ *w in P^fs^) and y £ y* in BR (Sj), where s^, s^ run over A, our result follows
-30-Proposition 3.9 shows that for any X € the fibre o~1 (X) is a union of some P-eguivalence classes of An and then is also a union of some PL ~(resp. PR-) equivalence classes of An .
From Lemma 3.6, we shall ask whether there exists some element w € o_ 1 (X) for any X € An such that ir(<C(w)) - o(w)
(resp. ir (R (w) ) » a(w)). The answer is affirmative. First let us show the following result.
Lemma 3.10 For w € An , we assume that
E * {e(it ♦ qn, + qn) |1 < t < i, q € X} is a set of entry classes of w such that i 1 - n < i . < i.
1 l
-| K s e e < i 1 and * 1 “ n < < j * - 1 ^
e e e ^ 3 | * Let S = 1|1 < u < m} be
a descending chain of w with i' m < ... < M and j; > . • • > Then |E n S| < 1.
Pro o f : Otherwise, there exist o,6 with 1 < o < 6 < m such that 3a » Ja and 3^ = 3b . Then there exist p,q € S such that
- ja + PH* - ia ♦ P.n, - jb + q.n and i£ - ifa + qn. So the inequality (ia~ib ) + (p-q)n < 0 follows from i^ - i£ < 0 and
(j -j. ) ♦ (p-q)n > 0 from j' - jl > 0, i.e. We have
a d a ts
| ia - ib < (q-p)n ( D
ja-jb > (q-p)n (2)
Since 0 < |ia"ib J» |ja-jb | < n ' the inequalities ( 1 ), (2) imply q-p > 0 and q-p < 0, respectively. Hence q ■ p and we have
r < 0 (3)
Fix a partition X * {X.j > X2 > •. • > Xr > € An * Let
X ,X be a permutation of X.,X,»...»X„. Assume that
a i a 2 ar
J = U ... U Jr € A With Jj = {saj+1'SUj+2',**'Saj+ a^-l*'1
r j
and
Ou =
i♦
Z X for some i €X.
Let w be the longest3 h* j + 1 *h
element in Wj. Then ir(f(wQJ )) * tt(8(woJ )) = n(J) “ X. As an affine matrix, w oJ has the form
r-1
where K j - ia a xaj * Xa^ dia9onal block of W QJ » 1 < j < r. The integer libelling the first row (resp. column) of K j in wQJ is aj+1. In other words, w 0J has the MDC form (Ar ,Ar-1,...,A 1) at 0 with the t-th entry e(i£r 3h* of Ah^W QJ ^ 1 < h < r , 1 < t < Xa^»
+ 1-t. where ij - ah +t, - ah _.,
Assume that p - {p., > p2 > > p. } is the dual partition A1
of X. Then it is clear that for any 1 < t < X ^ 4*
* 1
- n < i* < i* <
61 (t) M * *
V (t)
- n < j* < <
B^t) aB^(t)
v
(t)
... < i_ and
... < where
“ s ^ t )
ft,(t), 0 , (t),...,B (t) is the subsequence of 1,2,...,r such that
1 *
-
32-E. = ie(i^ + qji, j* ♦ qn) | 1 < h < y . , q € X } be the
r a Bh (t) Bh (t)
set of entry classes of w for 1 < t < X^. Then by Lemma 3.10,
the intersection of with any descending chain of w QJ has
cardinal at most 1. Assume that S ■ U ... U S^ is a disjoint union of k descending chains of w QJ satisfying Cn (w,k), k > 1. Then the cardinal of the intersection of E. with S is at most min {y.,k>, for any t with 1 < t < X.. Since U E is the full
t 1 t-1 r
set of entry classes of wQJ , it implies that
M k j
|S| < £ min {y.,k} * £ X.. Hence o(w ) < X. By Lemma 3.6,
t=1 z j«1 3
this implies that a(wQJ ) ■ if (£ (wQJ )) ■ ir(R(w0 J )) = X. So we have Lemma 3.11 For any J € A, let woJ be the longest element in Wj. Then we have
o(woJ ) ■ it (I (wQJ )) * ir(S(woJ )) ■ ir(J). d
Corollary 3.12 The map o»*n •* AR is surjective.
CHAPTER 4 : SOME CELLS OF THE AFFINE WEYL GROUP An
Let Cj.
{sr sr+1***«t > 8r ,8r ,_1 ...st |r < t, r' > t} for 1 < t < n and let C ■ U C. . In this chapter, we shall first
t«1 r
show that the sets {1}, C are both 2-sided cells of An correspond
ing to the partitions { 2 > 1 > . . . > 1 } € An »
respectively.
for any t, 1 < t < n. Then by applying this result to A2 , we We shall also show that C fc is a left cell of Ar
can find all cells of A2 .
Proposition 4.1 Let W be any Coxeter group. Then the set consisting of the identity element 1 is a 2-sided cell of W .
Proof: Otherwise, there exists some element w \ 1 such that w ? 1. So there exists a sequence of elements x„ ■ 1,
rW °
x . ,••«,X r - w with r > 1 such that for each j, 1 < j < r, either x^_1 -< x^ or x^ ^ x j_i» and either £ 1 x ^ , 1 * £(x^) or R(Xj_1) £ «(Xj). But this is impossible since £(xQ ) - R(xQ ) * 0 and then both £(xQ ) e jC(x.j) and R(xQ ) c S(x^). □
By Proposition 4.1, we see that {1} is a 2-sided cell of An .
Lemma 4.2 Let w € C. Then |£(w) 18 (w)
that either |£(w)| or |8(w)| is greater than 1. By symmetry, we may assume that |£(w)| > 2 and w = srsr+i***st with r < t-1. Then there exists u with u ? r such that i(suw) < i, (w). By the exchange condition, there exists m with r < m < t such that
su srsr + 1 * * *sm srsr-t>1 ** *sm + 1 ' So 1 (su srsr+1 * * ,sm + 1) < l(8r8r+1***
sm + 1 >. By assumption on i(w), we must have w = srsr + i•••sm + v We claim u r ~ F T . Otherwise, srsu s
r*1* * *8m 8r8r+1* *'sm+1 and it follows that su sr + i***sm s ^ , • • •S-.j.i • r+i in+ 1 Hence w ’ sr+1sr+2* * * s € C with i,(w') < i(w) and |jC(w')| > 2. This contradicts our hypothesis. If u « r-1, then sr_i8r *•*sm “ srsr+1* **8m + 1 * By symmetry, we have 8r 8r+1••-8m+1 = ®r_v 8r_v + 1 ••-sm+1.v for any v, 1 < v < n. Thus £ (w) = A. This is impossible by Lemma 2.2.5.
i . e . If u - r+T, then 8r + i8r8r+18r+2 * **sm “ 8r 8r+1* **8m + 1 *
8r8r+18r8r+2‘ * *sm 8rsr+1* *,sm + 1 * It follows that
8r8r + 2 8r+3* * *8m “ 8r+2***8m + ,r So w " " 8r+28r+3** *8m+1 € C satisfies U w " ) < i(w) and |£(w")| > 2. This also contradicts our hypothesis. So our result follows.
Lemma 4.3 (i) Cfc lies in some left cell of for any t, 1 < t < n. (ii) C lies in some 2-sided cell of *n
P r o o f : Let w - 8r8r + i***8t e ct for 8ome r, r < t. Then by Lemma 4.2, £ (w) -
{sr>,
£ <»r+18r+2‘••*t) “ {8r+1J * So we haV8 w L *r+18r+2***8t* Similarly» w* havewe can show that srisri_i • **st £ st for any r', r' > t. Therefore, any element of Ct lies In the same left cell as st> (1) Is proved.
Also, we cam easily check that sfc £ st st+1 L st+1 for any t, 1 < t < n. So A lies In some 2-sided cell of AR . By the proof of (1), any element of C lies in the same left cell as some s € A. This implies that C is contained in some 2-sided
cell of a
n
To show that C is a 2-sided cell of &n , we need the following result which appears in [1, (2.3e), (2.3f)].
Lemma 4.4 Assume that W is a Coxeter group, x,y € W and s is a Coxeter generator of W .
(i) If x < y, sy < y, sx > x, then x -< y if and only if y * s x . (ii) If x < y, ys < y, xs > x, then x ■< y if and only if y = xs. o
Proposition 4.5 (i) C is a 2-sided cell of An . (ii) Ct is a left cell of Ar for any t, 1 < t < n. (ill) C is a union of n left (resp. right) cells of AR .
Proof t (i) It is enough to show that if x € C, y t C, then
x J* y. By Proposition 4.1, we may assume that y |l 1. If x ^ y,
1W W
36
since otherwise we would have y' € C and this contradicts our hypothesis. Without loss of generality, we may assume that x' = sr sr + i***st for some r < t. We claim s / sr± v For otherwise, y' must have one of the following forms:
sfc with r < t
But in cases (1), ( 2), we have y' £ C| in case (3), when n * 2, y' e C, when n > 2, JC(y') = {sr > “ f(x'). All these cases contradict our hypothesis. Then s + sr±i* This implies that y' ■ sr ssr+1***st 4,1,3 hence £(y'). i.e. £(x') c I(y'). This still contradicts our assumption. So we have proved (i).
(ii) For any t, t' with t / t' and for any y € C fc, w € Cfcl, we have 8(y) ■ {sfc} + {st ,> - 8(w). By Theorem A(i), this
implies that y ^ w and hence by Lemma 4.3, and Ct , belong to
L n
the different left cells of A . But by (i), C ■ U C. is a
n t-1 r
2-sided cell of A_ which must be a union of some left cells of n
An . Zt turns out that Ct is a left cell of An for any t, 1 < t < n. (iii) By (i), (ii), we see that C is a union of n left cells of An . Since the map w -► w _1 in Afi induces a bijection between the set of left cells of A_ and the set of right cells of A . and
n n
since C is invariant under this map, this implies that C is also a union of n right cells of AR . o
Proposition 4.6 In the affine Weyl group A 2 , there are two 2-sided cells: {1} and C, where C consists of non-identity elements of A.» There are two left (resp. right) cells in C.
Proofs Note that A2 is actually am infinite dihedral group and that every non-identity element has one of the following forms:
(i) s 1 (s2s 1)*' (ii) s2 (s1s2 )*'
(iii) (s2s i )m (iv) (sls2 )m
where ¿ > 0 , m > 0 . Our result follows easily from Proposition
4.5. a
Proposition 4.7 Let X = {2 > 1 » ... > 1}., y ■ {1 > ... > 1} € A n .
Then {1} » a 1 (u ) and c g q 1 (X) .
P r o o f s First we shall show that {1) - o” *1 (y). It is clear that {1) c o“ 1 (y). On the other hand, if w + 1, let w ■ sty with
s t € A and i. (w) - 4(st ) + 4(y). By Lemma 3.4, we have o(w) > o(«t ). But it is easily seen that o(sfc) ■ {2 > 1 > ... > 1 ) . So w f o- 1 (p). Therefore (1) ■ o- 1 (y).
Secondly, we shall show that C ■ o~1 (X). When n * 2, we have A2 ■ {X,y>. It follows from {1) - a- 1 (y) and A 2 - {1} U C that C - o- 1 (X). Now assume n > 3. Then by the proof of Lemma 4.3, we see that C is actually a P-equivalence class of An . So b y Proposition 3.9, this implies that C c o_ 1 (X') for some X* € An < But we have s 1 € C and o(s.,) - X. Thus X' ■ X and then C c o~1 (X). Suppose w t C. Then there exists a reduced form w ■ s, s. ...i,
12 1t
-38-(i) 3jr 1 < j < such that I., j* i..^ ± 1
(ii) t > 3 and 3j, 1 < j < t, such that i j _ 1 * + 1 and TT =
In case (i), we have a(w) > o(s. s. ) = { 2 > 2 > 1 > . . . > 1 } > X. 4j*1
In case (ii), we have a (w) > a(s, s, s, )= {3 > 1 > ... > 1 } > X. lj-1 ij ij*1
So in both cases, w fL o_1 (X) . This implies that C = o- 1 (X). o Remark 4.8 We can show that when w € C, the length function M w ) has the much simplier form:
M w ) - (t)w - (t+1)w - [(t!w
OPERATIONS ON BLOCKS
In the remainder of our thesis, we shall always assume n > 3. We wish to determine the left cell of AR containing a given element w. We know from Theorem D that each PL -equivalence class of A lies in some left cell of A . But any two elements of a
n n
P -equivalence class of A can be transformed from one to another
L n
by a succession of left star operations. It will turn out that we can perform various interchanging operations on blocks of w which are successions of left star operations and so give us elements in the same left cell as w. Although we cannot in general obtain all elements in the same left cell as w in this way, the interchanging operation on blocks will be crucial in our subsequent determination of the left cells.
15.1 ITERATED STAR OPERATIONS
Assume that w £ A. has a DC form (A) at i £ Z with n
|A(w)| ■ r, 1 < r < n. Assume that the entry e(i,j(w)) of w satisfies j(w) < j^(w), where e((w), j£(w)) is the h-th entry of A(w) for 1 < h < r. Let k - max {h|1 < h < r, j^(w) > j(w)}. Then there exists a sequence of elements xQ » w, x.,...,xr_ 1, in AR such that for every 1 s t < r-1, we have xt ■ *xt_,j in (■¿•►t-l * * In particular, w' ■ x r-1 has a DC form (A) at i-1 with |A(w')|» r such that
u
4 0
-where e((w'), j^(w')) is the h-th entry of A(w') for 1 < h < r. and the entry e(i+r, j(w')) of w' satisfies j(w') «
Definition 5.1.1 For w, w' € An , 1 < r < n and i 6 X, we write w , ^ w , if (i) w has a DC form (A) at i with |A(w) | = r
(ii) 3 a sequence of elements xQ ■ w, = w' in Ar
such that for every 1 < h < r-1, we have x^ = *xh-1 in ®L^s i+h-1^ For w, w' € *n , i € X and 1 < r < r+m < n, we write
w' < — w, if 3 a sequence of elements xQ = w,
x , ,...,x_ - w' in *. such that for every 1 < h < m, we have
l m n
xh <*(^2-h,r_). we write w w . if w <*d+1-™,r,
Here is an example for w € Ar which has DC form (A) at i € I m)
w'
with |A(w)| - 3.
jlvd-A column
iJ(W) - 'di Column
Remark 5.1.2 (i) If w has a DC form (A) at i € X with |A(w)| = r < n, then we can easily check that w' with
*
w' <— w exists if and only if jQ < j^(w) • Also, w"
*
with w — w " exists if and only if j**(w) < where e(i,jQ ), e(i+r+1,jr + 1 ) are the entries of w.
(ii) Given w € A n , i € S, r , m > 0 with r + m < n, an element w 1 w* <— (l* 1'r.fm) w or w — ii ±lt£iJSL» w > is unique if it exists.
(iii) In general, the expression w — >r rm ^ > w ' is equivalent
to w < * „• but not to w* < * w .
§5.2 SOME RESULTS ON ITERATED STAR OPERATIONS
In this section, the first two lemmas give a necessary and sufficient condition on an element w of A„ for which the iterated
n
star operations on w can be carried out. The last lemma states a property of the iterated star operations.
Lemma 5.2.1 ■ ■ ■ ■■ Assume that w C A_ has a DC form (A) at i € S.n Let m satisfy 1 «< m < n-|A|. Then there exists w' with
w — tAl.1 1 w* if and only if A(w) is a longest descending
chain in A 1(w), where A*(x) is the block of x consisting of rows from the (i+1)-th to the (i+ |A| ♦ m)-th for x € Aft.
Proof t («) We apply induction on m > 1. The result is obvious for m ■ 1. When m > 1, since A(w) is a longest descending chain in the block [A(w), B(w)], where B(w) is the (i ♦ |AI ♦ 1)-th
*
